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[2] H. K. Chang and J. L. Liou, “An image encryption scheme based on quadtree compression scheme,” in Proc. Int. Computer Symp., vol. 1, pp. 230–237, 1994. [3] D. E. R. Denning, Cryptography and Data Security. Reading, MA: Addison-Wesley, 1983. [4] W. Diffie and M. Hellman, “New directions in cryptography,” IEEE Trans. Inform. Theory, vol. IT-22, pp. 644–654, Nov. 1976. [5] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston, MA: Kluwer, 1993. [6] C. J. Kuo, “Novel image encryption technique and its application in progressive transmission,” J. Electron. Imag., vol. 2, pp. 345–351, 1993. [7] C. Schwartz, “A new graphical method for encryption of computer data,” Cryptologia, vol. 15, pp. 43–46, 1991.
A Deblocking Technique for Block-Transform Compressed Image Using Wavelet Transform Modulus Maxima Tai-Chiu Hsung, Daniel Pak-Kong Lun, and Wan-Chi Siu
Fig. 6. Decrypted image (PSNR
= 31.36 dB) in our second experiment.
original image. Since the distortion between V and V c is limited, V c is also significant. Illegal users cannot detect that V c is fake without any hint, even if they steal V c : On the other hand, suppose the illegal users detect that V c is fake, and want to break it. Then the outlaws may apply the following three types of attacks to break our cryptosystem. The first is the ciphertext-only attack [3]. In this attack, the illegal users are assumed to have only a cipherimage V c ; and do not have the private key k: The outlaws cannot obtain w; h; no ; G; and D in this case since these parameters are encrypted by the DES-like method and k: Thus, the thieves cannot obtain our original image. However, suppose the illegal users try to guess k by brute force. Let the bitsize of k be 112. Then, k has 2112 possible combinations. Here we assume that our cryptosystem employs double DES to encrypt w; h; no ; G; and D: A private key has 56 bits in DES. So, the key k has 112 bits in double DES. If the illegal users employ a 100 MIPS computer to conjecture k; the computational load is then 112 6 18 2 =(100 2 10 2 60 2 60 2 24 2 365) = 1:646 2 10 years. This is, indeed, a very long time. Thus, our cryptosystem is secure for ciphertext-only attack. The other two attacks are the known-plaintext and chosen-text attacks [3]. They are more religious than the ciphertext-only attack. In these two attacks, the illegal users are assumed to have obtained several plainimage and cipherimage pairs, and all of these pairs share a common key k: In these cases, the outlaws can analyze these pairs to obtain the common key k; and correctly decrypt the next cipherimage if the sender still encrypts his next original image by k: To prevent these attacks, we define that our private key is disposable, i.e., it is a one-time pad system [3]. Since no common key exists in our cryptosystem, no one can break our cryptosystem by the known-plaintext or the chosen-text attack. REFERENCES [1] N. Bourbakis and C. Alexopoulos, “Picture data encryption using scan patterns,” Pattern Recognit., vol. 25, pp. 567–581, 1992.
Abstract— In this work, we introduce a deblocking algorithm for Joint Photographic Experts Group (JPEG) decoded images using the wavelet transform modulus maxima (WTMM) representation. Under the WTMM representation, we can characterize the blocking effect of a JPEG decoded image as: 1) small modulus maxima at block boundaries over smooth regions; 2) noise or irregular structures near strong edges; and 3) corrupted edges across block boundaries. The WTMM representation not only provides characterization of the blocking effect, but also enables simple and local operations to reduce the adverse effect due to this problem. The proposed algorithm first performs a segmentation on a JPEG decoded image to identify the texture regions by noting that their WTMM have small variation in regularity. We do not process the modulus maxima of these regions, to avoid the image texture being “oversmoothed” by the algorithm. Then, the singularities in the remaining regions of the blocky image and the small modulus maxima at block boundaries are removed. We link up the corrupted edges, and regularize the phase of modulus maxima as well as the magnitude of strong edges. Finally, the image is reconstructed using the projection onto convex set (POCS) technique [2] on the processed WTMM of that JPEG decoded image. This simple algorithm improves the quality of a JPEG decoded image in the senses of signal-to-noise ratio (SNR) as well as visual quality. We also compare the performance of our algorithm to the previous approaches, such as CLS and POCS methods. The most remarkable advantage of the WTMM deblocking algorithm is that we can directly process the edges and texture of an image using its WTMM representation. Index Terms—Image enhancement, wavelet transforms.
I. INTRODUCTION Transform coding is an efficient block-based image compression technique that has been widely used in the image compression industry. In particular, the discrete cosine transform (DCT) has been adopted as the basic compression algorithm of the Joint Photographic Experts Group (JPEG), Motion Picture Experts Group (MPEG), and others. For conventional transform coding, an image is first divided into a number of n 2 n nonoverlapped blocks. Each block is Manuscript received June 6, 1996; revised August 12, 1997. This work was supported by Hong Kong Polytechnic University under Grant 340/808/A3/420. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Moncef Gabbouj. The authors are with the Department of Electronic Engineering, Hong Kong Polytechnic University, Hung Hom, Hong Kong (e-mail:
[email protected]). Publisher Item Identifier S 1057-7149(98)06811-0.
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Fig. 1. Effect of WTMM thresholding and denoising near edges of blocky Lena image; see Sections III-A and III-B.
transformed using the DCT and followed by quantization and variable length coding. However, in such a block-based transform coding scheme, the correlation among adjacent blocks is not exploited. Due to the quantization errors generated in different blocks, reconstructed images exhibit blocking artifacts along block boundaries when the compression rate is high. Such an artifact, the so-called blocking effect, is considered to be the most disturbing artifact in the reconstructed images. Researchers have suggested various methods to tackle this problem. They include the lowpass filtering method [3], constrained optimization method [4]–[6], the projection onto convex set technique [7], and neural network method [8]. They use different optimization cost functions and constraints so as to limit the variation of block boundaries, etc. These methods indeed reduce the blocking effect in the sense of improving the signal-to-noise ratio (SNR). However, the visual quality of the processed image does not always appear to be much improved. This is because the visual importance is not included in the cost function of some optimization methods for deblocking. Another approach of deblocking is by wavelet shrinkage [9]. This method first transforms the blocky image into the wavelet domain, and performs the soft thresholding technique [10] on the wavelet coefficients. The image is then reconstructed by using the inverse wavelet transform. The wavelet shrinkage approach has an advantage over the other deblocking approaches in that it effectively removes noise and preserves edges, which are of great visual importance to human beings. However, the blocking effect problem cannot be solved simply by denoising, since strong irregularity is found on the block boundary. Furthermore, this method may oversmooth the image over some texture regions such as the hair in the baboon image and in the Lena image, since these regions behave similar to noise. Recently, the wavelet transform modulus maxima (WTMM) representation was used to characterize a signal based on the Lipschitz exponents [1]. It enables local and effective operations on multiscale edges. With the WTMM representation, the blocking effect of a JPEG decoded image can be characterized as: 1) small modulus maxima at block boundaries over smooth regions; 2) noises or irregular structures near strong edges; and 3) corrupted edges across block boundaries. We then develop a deblocking algorithm based on these observations. In the next section, we first briefly review the WTMM and its reconstruction. Then, we describe our observations and
introduce a deblocking algorithm under the WTMM representation. The proposed algorithm shares the same advantage as the wavelet shrinkage approach that the edges of the image, which are sensitive to human beings, will not be affected during the operation. It is different from the previous approaches in that it allows one to localize the deblocking operation onto the block boundaries and the irregular structures of the image only. Hence, regions other than those will have less influence. Deblocking results using the proposed WTMM deblocking algorithm are given at the end of this work and are compared with some existing deblocking methods. The enhancement of the image in using the proposed approach is clearly demonstrated. The proposed algorithm improves the quality of JPEG decoded images in the senses of signal-to-noise ratio (SNR) as well as visual quality. This makes the current algorithm different from the wavelet shrinking or the projection onto convex set (POCS) methods, which sometimes may not give significant improvement in visual quality. II. WAVELET TRANSFORM MODULUS MAXIMA REPRESENTATION The discrete dyadic wavelet transform [1], [2] of an image
f is defined as fS2 f; (W21 f )1j J ; (W22 f )1j J g where f 2 L2 (R2 ): Each component is obtained by the convolutions of f (x; y ) with the scaling function and the dilated wavelets: S2 f = f 3 2 (x; y ); W21 f (x; y ) = f 3 21 (x; y ); W22 f (x; y) = f 3 22 (x; y): The wavelets are designed to be the partial derivatives of a smooth function along x-direction and y-direction, respectively. That’s, 1 (x; y) = @ 8(x; y)=@x and 2 (x; y) = @ 8(x; y)=@y: Denote also the modulus of wavelet transform M2 f (x; y ) = jW21 f (x; y)j2 + jW22 f (x; y)j2 and the phase A2 f (x; y ) = arctan(W22 f (x; y )=W21 f (x; y )): For each scale 2j ; a particular point (x; y ) is detected as a WTMM if M2 f (x; y ) > M2 f (x1 ; y1 ) and M2 f (x; y ) > M2 f (x2 ; y2 ) where (x1 ; y1 ); (x2 ; y2 ) are the adjacent locations in the direction indicated by the phase A2 f (x; y ) in eight quantized directions: 0; =4; =2; 3=4; ; 0=4; 0=2; 03=4: The WTMM representation records the positions of modulus maxima ((xj;u ; yj;u ))u2Z as well as the values M2 f (xj;u ; yj;u ) and A2 f (xj;u ; yj;u ) for levels 1 j 2J and the smoothed image S2 f (x; y ): In this work, the realization of the dyadic wavelet transform follows with that of Mallat’s approach [1], [2]. The image is first symmetrically
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(a)
(b)
(c)
(d)
Fig. 2. Effect of phase and amplitude regularization, see Section III-C. (a) Reconstructed Lena without phase and amplitude regularization. (b) Zoom-in of (a). (c) Reconstructed Lena with complete WTMM deblocking. (d) Zoom-in of (c).
extended to prevent the discontinuity induced near image boundaries. It is certainly possible to design another type of wavelet such that the regularity of the image can be more accurately estimated and free from the boundary effect. One of them is the interval wavelet. We refer readers to [11] for further details of the design of interval wavelet. III. CHARACTERIZATION OF BLOCKING EFFECT USING WTMM REPRESENTATION The blocking effect, which is incurred by the quantization errors of the image blocks, corrupts the image in several ways. For smooth regions, it may introduce small discontinuities at block boundaries with an average magnitude equals half of the quantization step. If there are edges inside or across several blocks, it would also introduce “ringing” artifacts and corrupted edges can be found near block boundaries of the image. Under the WTMM representation, we can characterize these artifacts and perform the corresponding corrections. Let us study such observations from wavelet reasoning.
A. Singularities Introduction over Smooth Region It is known that the position where the WTMM is found corresponds to the part of image where fast variation in amplitude is found [1], [2]. Hence, for a smooth region, there should be no modulus maxima residing on the first few levels, and so the detection of modulus maxima on block boundary over smooth region implies that they should be introduced by the blocking effect. Let K L2 (R2 ) be a local region of an image f (x; y ) 2 L2 (R2 ), which includes some blocks fi;j inside. The (i; j ) block of the image f is fi;j (x0 ; y 0 ): Then the decompressed image can be written as
^i;j (x ; y f 0
0
0
) = fi;j (x
0
= fi;j (x
;y ;y
0
) + I DC T [ei;j (u; v )]
0
)+e ^i;j (x
0
;y
0
)
where ei;j (u; v ) is the quantization error of the image block fi;j under the DCT domain. Let Kb to be the collection of blocks in K:
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(a)
(b)
(c)
(d)
Fig. 3. Locations of the WTMM with and without linking the disconnected WTMM; see Section III-D. (a) Processed WTMM at level 21 without linking. (b) Processed WTMM at level 21 with complete WTMM deblocking. (c) Processed WTMM at level 22 without linking. (d) Processed WTMM at level 22 with complete WTMM deblocking.
Taking the wavelet transform, we have ^(x; y ) W2 f
= W2
f (x; y )
+
average of the inverse DCT of the quantization error for lowfrequency DCT coefficients.
2
W2 e ^i;j (x; y )
(i;j ) K
2
W2 e ^i;j (x; y )
(i;j ) K
for l = 0; 1 1 1 ; Js ; (x; y ) 2 K: The expected magnitude of the WTMM detected at block boundary should then be equal to the
E (M
2
^(x; y )) E (M f
2
e ^i;j (x; y ))
(1)
for ei;j 2 Kb and (x; y ) 2 K: A simple thresholding on block boundary should discriminate most of them and the threshold should be valid for the same quantization table and can be estimated empirically.
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(a) Fig. 4.
(b)
“Nondeblocking” maps for Lena, see Section III-E. (a) Nondeblocking map at scale 21 : (b) Nondeblocking map at scale 22 :
TABLE I COMPARISON OF THE PERFORMANCE OF WTMM DEBLOCKING WITH OTHER CONVENTIONAL METHODS FOR JPEG-ENCODED IMAGES THAT ARE COMPRESSED WITH QUANTIZATION TABLE SHOWN IN TABLE II
B. Ringing Artifacts Near Edges Another artifact caused by the aforementioned quantization error is the ringing near edges. In a typical quantization table, more bits are allocated for low frequencies, hence the effect of quantization error increases for high-frequency contents of an image, such as edges. For strong edges, the effect of quantization error will be even more observable and becomes the ringing artifacts. Although it is very difficult to model such quantization error, we can still discriminate them under the WTMM representation using the denoising technique similar to [1] and [2], since the ringing behaves like noise near strong edges. The ringing artifact near edge is demonstrated in Fig. 1(a). To perform denoising, we first note the modulus maxima, which have a
decrease in magnitude as scale increases, or those that simply do not propagate to larger scales. These maxima are then removed explicitly from the WTMM representation of the image. Fig. 1(b) illustrates the result of WTMM thresholding and denoising near edges. We can see that most ringing artifacts near edges have been removed while the edges are preserved. C. Corruption of Edges When there are edges across block boundaries, the phase and magnitudes of their WTMM may be corrupted by the transform coding process. For such artifacts, we cannot remove them using thresholding or denoising. However, it is observed that the modulus
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(a)
(b)
(c)
(d)
Fig. 5. Results for Lena. (a) Original. (b) JPEG decoded, SNR 20.70 dB. (c) CLS deblocked, SNR 20.91 dB. (d) WTMM deblocked, SNR 21.40 dB.
maxima at the most detailed level of a corrupted edge have large variations in phase across scales and in magnitude from neighbor. By imposing a phase variation bound on these maxima across scales,
j
A2 f (x; y )
0
j
Af (x; y ) < Amax
TABLE II QUANTIZATION TABLE USED IN THE EXPERIMENT
(2)
where Af (x; y ) represents the mean phase across scale and Amax is the maximum phase change; the maxima that have large phase difference from the mean phase can then be selected and regularized. The procedure is as follows: first, we trace the WTMM across scales to estimate the mean orientation of these corrupted edges. Then, we have to check if the difference of phase satisfies (2) or not. If it violates (2), we set the phase of the current level maxima to the mean phase. After the phase regularization of such corrupted edges, we can then use the regularized phase to locate the neighboring WTMM along the tangential direction of the regularized phase or use the estimated Lipschitz exponents to estimate the original magnitudes.
By these regularizations, edges with small corruption can be readily corrected. However, further research is still required to correct those edges that are greatly corrupted. In Fig. 2, we show the effect of the phase and amplitude regularizations. We can see that there are some singular points that are regularized at block boundaries, such as the points near the hat.
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(b)
(c)
(d)
Fig. 6. Results for baboon. (a) Original. (b) JPEG decoded, SNR 15.77 dB. (c) CLS deblocked, SNR 15.88 dB. (d) WTMM deblocked, SNR 15.88 dB.
D. Linking of Disconnected Modulus Maxima Strong edges should survive in the denoising and thresholding processes, however, weak edges may be deleted when it is corrupted at block boundary, since it is small in amplitude. To save the weak edges from becoming dotted edges after processing, we link the disconnected modulus maxima which have the gap of only one pixel for the first few levels. For all 3 2 3 windows that do not have any modulus maxima at the center, we search for a pair of modulus maxima such that they share the same tangential direction through the center. If a pair of modulus maxima is found, a modulus maximum is added at the center with magnitude and phase to be the average of the found pair. In Fig. 3, we show the effect of linking the disconnected WTMM. E. Segmentation for WTMM Processing From the above processes, we can effectively reduce the blocking effect over smooth regions, reduce the ringing near edges, and regularize the corrupted edges. However, it is observed that the WTMM over texture regions may be wrongly classified as irreg-
ular structures and removed. For this problem, we need to use a segmentation technique to discriminate these texture regions to avoid “overdeblocking.” The blocking effect is relatively less observable for texture regions and the modulus maxima in these regions have only low local variation of regularity (but not no local variation) among the neighboring modulus maxima, so even if we do not perform any deblocking for the texture regions, the blocking effect will not significantly affect the visual quality of the overall image. To discriminate the maxima that are most probably texture, we suggest using the local variance of Lipschitz exponent for the classification. We refer to [1] for a detailed description of Lipschitz exponent. The local variation of Lipschitz exponent of a particular maximum located at x; y is defined as
( )
V2
1 ( ) = # max
3
x; y
u=
3
(2 ( +
03 v=03
l
x
u; y
+ ) 0 2 ( ))2 u
l
x; y
(3)
( ) ( )
where we denote the Lipschitz exponent of a WTMM located at x; y estimated at level j to be l2 x; y ; the local mean of l2 x; y to be l2 x; y ; and the number of WTMM inside the window to be
( )
2
( )
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(a)
(b)
(c)
(d)
Fig. 7. Results for peppers. (a) Original peppers. (b) JPEG decoded, SNR 21.94 dB. (c) CLS deblocked, SNR 22.15 dB. (d) WTMM deblocked, SNR 22.66 dB.
# max : Suppose we have computed s levels of WTMM of the blocky image, we can generate a “nondeblocking” map by the following segmentation algorithm.
( )
2
2
1) Compute V2 x; y of the blocky image for scale 1 to s using (3) with a 7 2 7 window. 2) For the irregular maxima, i.e., l2 x; y < ; set it to be “nondeblocking” maxima if V2low < V2 x; y < V2up :
( ) 0 ( )
Then, we have a map for indicating the locations where no deblocking is required. Fig. 4 shows the “nondeblocking” map for Lena, we can see that the modulus maxima near the hair have been marked while the maxima due to ringing, noises and the modulus maxima at block boundary are not selected. The parameters V2low ; V2up can be obtained empirically, and it works for all images.
IV. THE WTMM DEBLOCKING ALGORITHM Collecting the techniques of blocking artifacts reduction, ringings and noise elimination, regularization and segmentation on the
WTMM representation of an image, we propose a novel WTMM deblocking algorithm as follows. 1) Compute s levels of WTMM of the blocky image, chain maxima curves across scales and estimate the Lipschitz exponent for each maxima chain. 2) Segmentation: generate nondeblocking map using the method as stated in Section III-E. 3) Denoising: remove maxima with the Lipschitz exponents smaller than a threshold 1 < ; which implies that either the magnitudes of these maxima decrease as the scale increases or these maxima simply do not propagate to larger scales. 4) Thresholding: for scales 1 to s ; remove maxima located at block boundaries with amplitudes below a threshold 2 : 5) Regularization: set the phase of WTMM to the mean phase along scales if it violates (2), then average the amplitude of strong edges at block boundaries using the WTMM along tangential direction of the regularized phase with a selected length 3 .
0
2
2
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6) Linking: link unconnected modulus maxima for the scales 1 to s . 7) Reconstruction: reconstruct the image from the processed WTMM using the POCS as in [2]. In our experiments, we use a three-level WTMM representation, s : We select to process the first two levels of the WTMM s ; since no blocking effect is visible in the WTMM at levels higher than two. Furthermore, it saves computational effort in realization. ; V2up ; V2low ; The other parameters used are V2low up ; 1 0 : ; 2 : ms where ms is the maximum V2 values of WTMM at scale s; and Amax : ; 3 : After processing the WTMM, the deblocked image is reconstructed by using the POCS method. We apply the proposed algorithm for the deblocking of some test images including Lena, baboon, peppers, etc.; improved results are obtained in the senses of both SNR and visual quality. Fig. 5(a) shows the original Lena and Fig. 5(b) shows the JPEG decoded Lena. We show the deblocked result using the constraint least square method [4] in Fig. 5(c) and the proposed WTMM deblocked result of Lena in Fig. 5(d). Results for baboon and peppers are also included in Figs. 6 and 7. In the experiments, we use the same set of parameters for deblocking the test images since the quantization tables used in generating the test images are the same. Table I shows a comparison of the performance of WTMM deblocking with other conventional methods for JPEGencoded images that are compressed with quantization table shown in Fig. 2. The methods are 1) nonlinear interpolative decoding NID0 [12], 2) constraint least square CLS [4], 3) projection onto convex set method: P OCSRZ from [5] and P OCSY GK from [4]. We denote W T MM to be the WTMM deblocking method, W T MMnpr to be the WTMM deblocking method without regularization. We also denote bpp to be the number of bits per pixel, SNR to be the difference of the deblocked image in SNR from the original JPEG decoded image. It is seen in the table that the WTMM deblocking algorithm gives a consistent improvement in SNR for all test images. Furthermore, the improvement is the largest for the proposed algorithm as compared with other approaches.
2
= ( = 1 2)
3
= 900
= 0 08
= 0 11
(
=5
= 200 = 10 = 0 08 =3
(
)
[3] H. C. Reeve and J. S. Lim, “Reduction of blocking effect in image coding,” Opt. Eng., vol. 23, pp. 34–37, Jan. 1984. [4] Y. Yang, N. P. Galatsanos, and A. K. Katsaggelos, “Regularized reconstruction to reduce blocking artifacts of block discrete cosine transform compressed images,” IEEE Trans. Circuits Syst. Video Technol., vol. 3, pp. 421–432, Dec. 1993. [5] R. Rosenholtz and A. Zakhor, “Iterative procedures for reduction of blocking effects in transform image coding,” IEEE Trans. Circuits Syst. Video Technol., vol. 2, pp. 91–95, Mar. 1992. [6] S. Minami and A. Zakhor, “An optimization approach for removing blocking effects in transform coding,” IEEE Trans. Circuits Syst. Video Technol., vol. 5, pp. 74–82, Apr. 1995. [7] Y. Yang, N. P. Galatsanos, and A. K. Katsaggelos, “Projection-based spatially adaptive reconstruction of block-transform compressed images,” IEEE Trans. Image Processing, vol. 4, pp. 896–908, July 1995. [8] S.-W. Hong, Y.-H. Chan, and W.-C. Siu, “The neural network modeled POCS method for removing blocking effect,” in IEEE Int. Conf. Neural Networks (ICNN’95), vol. III, pp. 1422–1425, Nov. 1995. [9] R. A. Gopinath, H. Guo, M. Lang, and J. E. Odegard, “Wavelet-based post-processing of low bit rate transform coded images,” in Proc. 1994 IEEE Int. Conf. Image Processing (ICIP’1994), vol. II, pp. 913–917, 1994. [10] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, vol. 81, pp. 425–455, 1994. [11] A. Cohen and I. Daubechies, “Wavelets on the interval and fast wavelet transforms,” Appl. Computat. Harmonic Anal., vol. 1, pp. 54–81, 1993. [12] S. W. Wu and A. Gersho, “Nonlinear interpolative decoding of standard transform coded images and video,” in Proc. SPIE Conf. Image Processing Algorithms and Techniques III, vol. 1657, pp. 88–99, 1992.
)
1
V. CONCLUSION In this correspondence, we proposed a local deblocking algorithm for JPEG decoded images using the wavelet transform modulus maxima (WTMM) representation. The proposed WTMM deblocking algorithm first performs a segmentation to identify the texture regions by noting that their WTMM would have the Lipschitz exponents with low local variances. We do not process these maxima to avoid the image texture being “oversmooth” by the algorithm. Then, thresholding, denoising, phase and amplitude regularization are performed for other maxima, and the corrupted edges are linked up. Finally, we reconstruct images from the processed WTMM using the POCS method. The resulting deblocked image is improved in terms of visual quality and signal to noise ratio as compared with other deblocking algorithms such as the CLS approach. REFERENCES [1] S. Mallat and W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inform. Theory, vol. 38, pp. 617–643, 1992. [2] S. Mallat and S. Zhong, “Characterization of signals from multiscale edges,” IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 710–732, July 1992.
Quantization Error in Regular Grids: Triangular Pixels Behrooz Kamgar-Parsi and Behzad Kamgar-Parsi
Abstract—Quantization of the image plane into pixels results in the loss of the true location of features within pixels and introduces an error in any quantity computed from feature positions in the image. Here, we derive closed-form, analytic expressions for the error distribution function, the mean absolute error (MAE), and the mean square error (MSE) due to triangular tessellation, for differentiable functions of an arbitrary number of independently quantized points, using a linear approximation of the function. These quantities are essential in examining the intrinsic sensitivity of image processing algorithms. Square and hexagonal pixels were treated in previous papers. An interesting result is that for all DS < 1:13, where possible cases 0:99 < DT =D T and S are the MAE in triangular and square tessellations.
D
D
Index Terms—Quantization error, spatial quantization, triangular pixels, 2-D regular grids.
I. INTRODUCTION Digital processing of images requires quantizing (digitizing, discretizing) the image plane into pixels. This spatial quantization introduces an error in any quantity computed from spatial positions of features in the image, for example, in computing the length of a Manuscript received September 14, 1994; revised May 12, 1997. This work was sponsored by the Office of Naval Research. B. Kamgar-Parsi is with the Navy Center for Applied Research in Artificial Intelligence, Naval Research Laboratory, Washington, DC 20375 USA (e-mail
[email protected]). B. Kamgar-Parsi is with the Information Technology Division, Naval Research Laboratory, Washington, DC 20375 USA. Publisher Item Identifier S 1057-7149(98)06860-2.
1057–7149/98$10.00 1998 IEEE
